For REF , by [1]} \(\Vert \partial ^\alpha \tilde{g}\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\lesssim \Vert \tilde{g}\Vert _{{A}_{pq}^{s+m}(\mathbb {R}^n)}\)  holds for \(|\alpha |\le m\) . Note that \(\tilde{g}|_U=g\)  implies \(\partial ^\alpha \tilde{g}|_U=\partial ^\alpha g\) , thus taking restrictions we get \(\Vert \sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \Vert _{{A}_{pq}^s(U)}\lesssim \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{A}_{pq}^{s+m}(U)}\) . Therefore \(\Vert f\Vert _{{A}_{pq}^s(U)}\lesssim \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{A}_{pq}^{s+m}(U)}\)  whenever \(f=\sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \) .
Conversely for \(f\in {A}_{pq}^s(U)\) , take an extension \(\tilde{f}\in {A}_{pq}^s(\mathbb {R}^n)\) , we have \(\tilde{f}=(I-\Delta )(I-\Delta )^{-1}\tilde{f}=\tilde{g}_0+\sum _{j=1}^n\partial _j\tilde{g}_j\)  where \(\tilde{g}_0:=(I-\Delta )^{-1}\tilde{f}\)  and \(\tilde{g}_j:=-\partial _j(I-\Delta )^{-1}\tilde{f}\)  (\(1\le j\le n\) ). Thus by [1]} we have \(\sum _{j=0}^n\Vert \tilde{g}_j\Vert _{{A}_{pq}^{s+1}(\mathbb {R}^n)}\lesssim \Vert \tilde{f}\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\) . Taking infimum of all extensions \(\tilde{f}\)  over \(f\)  we get REF  for \(m=1\) . Doing this recursively we get all \(m\ge 1\) .
For REF , we have \(C^{k,s}(\mathbb {R}^n)={B}_{\infty \infty }^{k+s}(\mathbb {R}^n)\)  from [3]} for \(k\ge 0\)  and \(0<s<1\) . Thus in this case \({B}_{\infty \infty }^{k+s}(U)=\lbrace \tilde{f}|_U:\tilde{f}\in C^{k,s}(\mathbb {R}^n)\rbrace \subseteq C^{k,s}(U)\)  for arbitrary open subset \(U\) . When \(U=\mathbb {R}^n_+\)  or \(U\)  is bounded smooth, by the existence of extension operator \(C^{k,s}(U)\rightarrow C^{k,s}(\mathbb {R}^n)\)  (see [4]}) we get \(C^{k,s}(U)\subseteq {B}_{\infty \infty }^{k+s}(U)\) . This proves the case \(k\ge 0\) .
Comparing the result REF  and (REF ) we see that \(C^{k,s}(U)={B}_{\infty \infty }^{k+s}(U)\)  for \(k<0\)  as well.
See [5]} and [6]} for REF .
See [1]} and [8]} for REF .
See [9]} for REF . In the reference the collection \(\lbrace B(x,2^{-J}):x\in \mathbb {R}^n,J\in \mathbb {Z}\rbrace \)  is replaced by \(\lbrace 2Q_{J,v}:=2^{-J}(v+(-\frac{1}{2},\frac{3}{2})^n):J\in \mathbb {Z},v\in \mathbb {Z}^n\rbrace \) , but there is no difference between the proofs. Note that one cannot use \(\lbrace Q_{J,v}:=2^{-J}(v+(0,1)^n):J\in \mathbb {Z},v\in \mathbb {Z}^n\rbrace \)  because for \(J_0\in \mathbb {Z}\) , the set \(\lbrace x\in \mathbb {R}^n:x_k\in 2^{-J_0}\mathbb {Z},\ \exists \,1\le k\le n\rbrace \)  is not contained in the union \(\bigcup _{J\ge J_0;v\in \mathbb {Z}^n}Q_{J,v}\) .
